Journal of Lie Theory
Volume 14 (2004) 151–163
c 2004 Heldermann Verlag
Koszul Duality of Translation—and Zuckerman Functors
Steen Ryom-Hansen
Communicated by B. Ørsted
Abstract.
We review Koszul duality in representation theory of category O ,
especially we give a new presentation of the Koszul duality functor. Combining
this with work of Backelin, we show that the translation and Zuckerman functors
are Koszul dual to each other, thus verifying a conjecture of Bernstein, Frenkel
and Khovanov. Finally we use Koszul duality to give a short proof of the EnrightShelton equivalence.
1.
Introduction.
The definite exposition of Koszul duality in representation theory is the paper of
Beilinson, Ginzburg and Soergel [5]. The main theme of that paper is that the
category O of Bernstein, Gelfand and Gelfand is the module category of a Koszul
ring, from which many properties of O can be deduced very elegantly. They also
consider singular as well as parabolic versions of O and show that they are are
equivalent to each other under the Koszul duality functor. Two of the main tools
to obtain these results were the Zuckerman functors and the translation functors.
In a recent paper by Bernstein, Frenkel and Khovanov [6] these functors are
studied on certain simultaneously singular and parabolic subcategories of O , in
type A. They show, partially conjecture, that each one of them – in combination
with an equivalence of certain categories due to Enright-Shelton – can be used to
categorify the Temperley-Lieb algebra and also conjecture that these two pictures
should be Koszul dual to each other.
The purpose of this note is to show that the translation– and Zuckerman
functors are indeed Koszul dual to each other. By this we mean that both
admit graded versions and that these correspond under the Koszul duality functor.
Although this of course is one of the main philosophical points of [5], a rigorous
proof was never given.
We furthermore use the Koszul duality theory to give a simple proof of the
Enright-Shelton equivalence.
c Heldermann Verlag
ISSN 0949–5932 / $2.50 152
Ryom-Hansen
∗
It is a pleasure to thank Wolfgang Soergel for many useful discussions.
Most of this work was done while I was a research assistant at the University of
Freiburg, Germany.
Added: Recently C. Stroppel [11] has obtained a proof of the full BernsteinFrenkel-Khovanov conjectures, in part using results from the present paper.
2.
Preliminaries
In this section we will mostly recall some of the basic definitions and concepts from
the theory of Koszul duality, see also [5].
Let g be a complex semisimple Lie algebra containing a Borel algebra b
with Cartan part h and let O be the category of g-modules associated to it by
Bernstein, Gelfand and Gelfand. For λ ∈ h∗ integral but possibly singular, denote
by Oλ the subcategory of O consisting of modules of generalized central character
χλ – we refer to it as the singular category O . Let S be the set of simple reflections
corresponding to our data, and let Sλ be the subset consisting of those reflections
that fix λ under the dot operation. Then Sλ defines a parabolic subalgebra q(λ).
We define the parabolic category Oλ to consist of the q(λ)-finite objects in O . For
λ = 0 we omit the index, i.e. we write O0 = O , this should not cause confusion.
Let P denote the sum of all indecomposable projectives in O , hence P is
a projective generator of O . Analogously, we can construct projective generators
Pλ (resp. P λ ) of Oλ (resp. Oλ ). Let A = EndO P (resp. Aλ = EndOλ Pλ , Aλ =
EndOλ P λ ). By general principles, we can then identify O with Mod−A, the
category of finitely generated A-right modules (and analogously for Aλ , Aλ ).
Let T0λ : O → Oλ , Tλ0 : Oλ → O be the Jantzen translation functors
onto and out of the wall. Passing to Mod−A, T0λ corresponds to the functor
Mod−A → Mod−Aλ : M 7→ M ⊗A X
where X is the (A, Aλ ) bimodule X = HomO ( Pλ , T0λ P ). There is a similar
description of Tλ0 : Oλ → O .
Let τλ : O → Oλ be the parabolic truncation functor, by definition it takes
M ∈ O to its largest q(λ)-finite quotient. It is right exact and left adjoint to the
inclusion functor ιλ : Oλ → O , which is exact, and it thus takes projectives to
projectives. It even takes indecomposable projectives to indecomposable projectives. Its top degree left derived functor is the Zuckerman functor that takes a
module M ∈ O to its largest q(λ)-finite submodule.
On the Mod−A level, τλ is given by the tensor product with the (A, Aλ )bimodule Y = HomOλ ( P λ , τλ P ). However, from the above considerations we
deduce that Y = Aλ , the left A-structure coming from τλ and the right Aλ structure coming from the multiplication in Aλ .
Let us finally recall the definition of a Koszul ring.
∗
Supported by EPSRC grant M22536 and by the TMR-network algebraic Lie Theory
ERB-FMRX-CT97-0100.
Ryom-Hansen
153
L
Definition 2.1. LLet R = i≥0 Ri be a positively graded ring with R0 semisimple and put R+ = i>0 Ri . R is called Koszul if the right module R0 ∼
= R/R+ admits a graded projective resolution P • R0 such that P i is generated by its component Pii in degree i. The Koszul dual ring of R is defined as R! := Ext•R (R0 , R0 ).
The main point of [5] is now that all the rings appearing above can be given a
Koszul grading.
3.
Koszul duality.
The main purpose of this section is to give a new construction of the Koszul duality
functor.
Koszul duality is an equivalence at the level of derived categories between
the graded module categories of a Koszul ring and its dual. It exists under some
mild finiteness conditions. The concept appeared for the first time in the paper
[3] where it is shown that for any finite dimensional vector space V the derived
graded module categories of the symmetric algebra SV and the exterior algebra
V
V ∗ are equivalent. The argument used there carries over to general Koszul rings
and is the one used in [5].
We shall here present another approach to the Koszul duality functor, using
the language of differential graded algebras (DG-algebras). Although also the
papers [4] and [9] link DG-algebras with Koszul duality, we were rather inspired
by the construction of the localization functor in the book of Bernstein and Lunts
[7].
Let us start out by repeating the basic definitions.
Definition
A DG-algebra A = (A, d) is a graded associative algebra
L∞ 3.1.
i
A = i=−∞ A with a unit 1A ∈ A0 and an additive endomorphism d of degree
1, s.t.
d2 = 0
d(a · b) = da · b + (−1)deg(a) a · db
and d(1A ) = 0.
Definition 3.2.
A left module M =
L(M, dM ) iover a DG-algebra A = (A, d) is
a graded unitary right A-module M = ∞
i=−∞ M with an additive endomorphism
2
dM : M → M of degree 1, s.t. dM = 0 and
dM (am) = da · m + (−1)deg(a) a · dM m
Definition 3.3.
A right module M = (M,
a DG-algebra A = (A, d)
L∞dM ) over
i
is a graded unitary right A-module M = i=−∞ M with an additive endomorphism dM : M → M of degree 1, s.t. d2M = 0 and
dM (ma) = dM m · a + (−1)deg(m) m · da
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Ryom-Hansen
Unless otherwise stated, A will from now on be the ring EndO (P ) from the
former section.
By the selfduality theorem of Soergel [10] we know that A = Ext•O (L, L),
where L is the sum of all simples in O . This provides A with a grading which by
[10] or [5] is a Koszul grading.
We regard A as a DG-algebra A with A = A0 . At this stage we neglect
the grading on A, hence A just has one grading, the DG-algebra grading.
Let K • be the Koszul complex of A, see e.g. [5]. It is a projective resolution
of k (which corresponds to the sum of all simples in O ). In [5] K • is a resolution
of left A-modules of k ; we however prefer to modify it to a resolution of right
A-modules. Furthermore we change the sign of the indices so that the differential
has degree +1.
Let kw denote the simple A-module corresponding to Lw ∈ O and let Kw•
be L
a projective resolution of kw . The Koszul complex K • is then quasiisomorphic
•
to
λ Kw .
We can consider K • as a DG right module of the DG-algebra A, and as
such it is K-projective in the sense of [7]: the homotopy category of DG-modules
of A is simply the homotopy category of complexes of A-modules, hence any
quasiisomorphism from K • to an DG-module M is a homotopy isomorphism.
Given two DG-modules M, N of the (arbitrary) DG-algebra A, recall the
construction of the complex Hom •A (M, N ):
Hom nA (M, N ) := HomA (M, N [n])
df := dN f − (−1)n f dM , f ∈ Hom nA (M, N ).
If we perform this construction on K • , we obtain a DG-algebra
End •A (K • ) = Hom •A (K • , K • ),
the multiplication being given by composition of maps (notice that this construction does not involve the Koszul grading on A).
The signs match up to give K • the structure of an End •A (K • ) left DGmodule.
Let for any DG-algebra A, Db (Mod−A) denote the bounded derived category of right A-modules, see for example Bernstein-Lunts [7] for an exposition of
this theory.
Let Db (Mof −A) denote the bounded derived category of finitely generated
A-modules, and let Db (Mof −A) be the corresponding category of A-modules.
Let Db (Mof − End •A (K • )) be the subcategory of Db (Mof − End •A (K • )) consisting
of finitely generated End •A (K • )-modules in the algebra sense. Then we have
Lemma 3.4.
Db (Mof − End •A (K • )) is generated, in the sense of triangulated
categories, by the Hom •A (K • , Kw• )’s.
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Ryom-Hansen
Proof.
Since our ring Ais Koszul, the DG-algebras
End •A (K • ) and (Ext •A (k, k), d = 0)
are quasiisomorphic [5, Theorem 2.10.1] and hence their bounded derived categories are equivalent. Under this equivalence the End •A (K • )-module
Hom •A (K • , Kw• ) becomes the (Ext •A (k, k) , d = 0)-module (Ext •A (k, kw ) , d = 0).
Now a certain polynomial DG-algebra B with d = 0 is studied in chapter
11 of [7]. Corollary 11.1.5 there is the statement that Db (Mof −B) is generated
by B itself. The proof of this Corollary 11.1.5 relies on the differential being zero
for B and of the fact that finitely generated projective modules over a polynomial
algebra are free. We do not have this last property in our situation and should
therefore adjust the statement of the Lemma accordingly. Let us give a few details
on the modifications:
Let thus (M, dM ) be an object of Db (Mof −(Ext •A (k, k), d = 0)). Then we
have a triangle in that category of the form
Ker dM → M → M/ Ker dM →
But the first and the third terms have zero differential so we may assume that
dM = 0.
Let now
0 → P −n → P −n+1 → . . . → M → 0
be a projective graded resolution of M considered as Ext •A (k, k)-module. Defining
i
P ∈ Db (Mof −(Ext •A (k, k), d = 0)) as having k ’th DG-part ⊕i Pk−i
and differential
∼
from the resolution, we get a quasiisomorphism P = M induced by .
On the other hand the finitely generated, graded projectives over Ext •A (k, k)
are the direct sums of the Ext •A (k, kw )’s and we are done.
Now since K • is K-projective, it can be used in itself to calculate
RHom A (K • , M ) for an A-module M. But K • is an End •A (K • )- module, so we
get a functor:
RHom A (K • , −) : Db (Mod−A) → Db (Mod− End •A (K • ))
This might require a little consideration: one should check that
RHom A (K • , −) takes homotopies to homotopies, even when considered as a functor to the category of right End •A (K • )-modules, and likewise for quasiisomorphisms. For homotopies, one checks this by hand (after having consulted e.g.
Bernstein-Lunts [7] for the definition of the homotopy category of DG-modules).
For quasiisomorphisms one uses the standard argument: if f : M → N is a quasiisomorphism, then Hom •A (K • , C(f )) is acyclic (C(f ) denoting the cone of f ) and
so on: the action of End •A (K • ) is irrelevant for this argument.
Now we also have a functor in the other direction:
L
− ⊗End •A (K • ) K • : Db (Mod−A) ← Db (Mod− End •A (K • ))
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Ryom-Hansen
Recall that this construction involves the replacement of
N ∈ Db (Mod− End •A (K • ))
by a K-flat object (actually a K-projective object will do), to which it is quasiL
isomorphic. The action of A on K • then provides N ⊗End •A (K • ) K • with the
structure of DG-module over A:
d(n ⊗ ka) = d(n) ⊗ ka + (−1)deg(n) n ⊗ d(ka)
= d(n) ⊗ ka + (−1)deg(n) n ⊗ d(k)a
= d(n ⊗ k)a
since the differential of A is zero. Once again, one should here check that the
functor makes sense as a functor into the category of A-modules.
We can then prove the following Theorem:
Theorem 3.5.
The functor RHom A (K • , −) establishes an equivalence of the
triangulated categories Db (Mof −A) and Db (Mof − End •A (K • )). The inverse funcL
tor is − ⊗End •A (K • ) K • .
Proof.
Note first that Db (Mof −A) is generated by the Kw• , since they are
resolutions of the simple A-modules and short exact sequences O -modules induce
triangles in Db (Mod−A).
But then RHom A (K • , M ) belongs to Db (Mof − End •A (K • )) for any M ∈
D (Mof − End •A (K • )), since it clearly does so for each of the generators Kw• .
b
We furthermore see that RHom A (K • , M ) is a K-projective End •A (K • )module for any M in Db (Mof −A) since it clearly holds for Kw• and the Kprojectives form a triangulated subcategory of K(Mod− End •A (K • )), the homotopy category of End •A (K • )-modules.
We then obtain a morphism ψM in Db (Mod − A) in the following way.
L
ψM : RHom A (K • , M ) ⊗End •A (K • ) K • → M : f ⊗ k 7→ f (k)
One checks that ψM defines a natural transformation from the functor
L
RHom A (K • , −) ⊗End •A (K • ) K • to the identity functor Id. It is a quasiisomorphism
for M = Kw• , and thus for all M .
On the other hand we have for a K-projective End •A (K • )-module N the
canonical morphism φN given as follows:
L
φN : N → RHom A K • , N ⊗End •A (K • ) K •
ni ∈ N i 7→ ( kj ∈ K j 7→ ni ⊗ kj )
One readily sees that φN is a quasiisomorphism for N = Hom •A (K • , Kw• ).
But these objects generate Db (Mof − End •A (K • )) and we can argue as above.
Ryom-Hansen
157
Let us finally comment on the shifts [1]. It should here be noticed that
since we are working with right DG-modules, the appropriate definition of M[1]
for a DG-module M over a DG-algebra A is the following:
(M [1])i = M i+1 , dM [1] = −dM and m ◦ a = ma
where m ◦ a is the multiplication in M [1] while ma is the multiplication in M .
So unlike the left module situation, the A structure on M is here left unchanged.
One now checks that the functors commute with the shifts [1].
Now A has a Koszul grading so we may consider the category mof −A of
finitely generated, graded A-modules. Let us use a lower index to denote the
graded parts with respect to this Z-grading. This passes to the derived category
of graded DG-modules which we denote by Db (mof − A). It may be identified
with the derived category of mof −A and carries a twist h1i which we arrange the
following way:
M h1ii = Mi−1
Now the grading on K • induces a grading on End •A (K • ) as well; it satisfies
the rule
End •A (K • )i := end •A (K • , K • h−ii)
With this grading on End •A (K • ), we may consider its graded module category, which we denote Db (mof − End •A (K • )). Let us moreover denote the subcategory of this generated, with twists, by the graded summands of the DG-module
End •A (K • ) itself by
Db (hEnd •A (K • )i)
(It would have been more consistent with the earlier notation to write Db (mof −
End •A (K • )) for this category; we however prefer the above notation in order to
save space).
It is now straightforward to prove the following strengthening of the former
theorem:
Theorem 3.6.
The functor RHom A (K • , −) establishes an equivalence of the
triangulated categories Db (mof − A) and Db (hEnd •A (K • )i). It commutes with the
twists [1] and h1i.
Our next task will be to study the DG-algebra (Ext •A (K • ), d = 0). More
precisely, we are going to compare Db (hExt •A (K • ), d = 0i) with the standard
derived category of the category of finitely generated, graded modules over
Ext •A (k, k), i.e. Db (mof − Ext •A (k, k)). Here, we define the Z-grading on
(Ext •A (K • ), d = 0) to be the negative of the DG-grading.
An object of Db h(mof − Ext •A (k, k)) is a bounded complex
0 → P −n → P −n+1 → . . . → P0 → . . . → P m−1 → P m → 0
of projective graded Ext •A (k, k)-modules (so the differential has degree 0). Using
this, one constructs a DG-module P of the DG-algebra (Ext •A (k, k), d = 0) in the
following way:
0 → P −n h−ni → P −n+1 h−n + 1i → . . . → P 0 h0i → · · · P m hmi → 0.
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Ryom-Hansen
L n
In
other
words,
P
is
the
module
P
=
whose k ’th DG-piece
Pk =
nP
L i
L is
i
i Pk−i . We make P into a graded DG-module by the rule Pj =
i P−j . This
construction defines a functor
F : Db (mof − Ext •A (k, k)) → Db (mod−(Ext •A (k, k), d = 0))
since a graded homotopy between two morphisms f, g : P • → Q• of modules will
be mapped to a homotopy in the DG-category.
I will now argue that actually F is a functor into the category
Db (hExt •A (k, k), d = 0i). To see this note first that the Pi hii all lie in this category
since they are summands of (shifts of) (Ext •A (k, k), d = 0). On the other hand
F can be viewed as an iterated graded mapping cone construction in the DGsense, starting with the morphism P m−1 hmi → P m hmi, which is already a DGmorphism, and the claim follows. The graded mapping cone of a graded DGu
module morphism M → N is given by C(u) = N ⊕ M [1] as DG-module, and
reversing the degrees with respect to the Z-grading.
It is now clear that F commutes with the twists [1]. On the other hand F
takes a sequence of Ext •A (k, k)-modules isomorphic to a standard triangle
u
M → N → C(u) → M [1]
to a sequence of graded DG-modules isomorphic to
F (u)
F (M ) → F (N ) → F (C(u)) → F (M [1]).
But this is easily seen to be a standard triangle of DG-modules; in other
words F is a triangulated functor.
One furthermore checks that F is full and faithful and it is thus an
equivalence of triangulated categories once we have shown that the generators
(Ext •A (k, kw ), d = 0) lie in the image of F . But this is clear.
Since
M1 → M2 → M3 →
is a triangle if and only if
F (M1 ) → F (M2 ) → F (M3 ) →
is a triangle we deduce that the inverse functor G is triangulated as well.
Our next task will be to analyze the behavior of the twists h1i with respect
to F . Now apart from the change of the sign of the differentials, we obtain the
relation:
F (M h1i) = F (M )[−1]h−1i.
L k
On the other hand, let for a complex P • of graded Ext •A (k, k)-modules P =
Pn
be the total module. We may then consider the map σ on P defined as follows:
p ∈ Pnk 7→ (−1)n p.
One now checks that σ defines an isomorphism between the functors F (−h1i) and
F (−)[−1]h−1i.
Ryom-Hansen
159
Recall once again that since Ais Koszul, the DG-algebras
End •A (K • ) and (Ext •A (k, k), d = 0)
are quasiisomorphic so there is an equivalence
φ : Db (Mod− End •A (K • )) → Db (Mod−(Ext •A (k, k), d = 0)) .
A closer look at the above quoted proof reveals that the quasiisomorphism is even
a graded one so we obtain an equivalence
φ : Db (hEnd •A (K • )i) → Db (hExt •A (k, k), d = 0i)
Let us gather the results in one theorem.
Theorem 3.7.
The composition of the above functors
G ◦ φ ◦ RHom A (K • , −) : Db (mof −A) → Db (mof − Ext •A (k, k))
is an equivalence of triangulated categories: the Koszul duality functor. It commutes with [1] and satisfies the rule
F (M h1i) = F (M )[−1]h−1i.
4.
Translation and Zuckerman functors.
Let us return to the translation functors to and from the wall T0λ : O → Oλ ,
Tλ0 : O → Oλ . We saw in the first section, that they can be viewed as functors
between the categories Mod −A and Mod −Aλ . Now, in [2] it is shown that they
can be lifted to graded functors
T0λ : mod −A → mod −Aλ
Tλ0 : mod −Aλ → mod −A
that still are adjoint and exact and such that T0λ takes pure objects of weight n to
pure objects of the same weight; this construction is based on Soergel’s theory of
modules over the coinvariants of the Weyl group. See also [1], where graded translation functors are constructed in the setting of modular representation theory.
The graded translation functor T0λ induces a homomorphism of graded DGalgebras End •A (K • ) → End •Aλ (T0λ K • ). But T0λ and Tλ0 are exact and K • is a
projective graded resolution of the pure module k , so T0λ K • is a projective graded
resolution of kλ . Hence T0λ K • is graded homotopic to the Koszul complex Kλ•
of Aλ and this implies that End •Aλ (T0λ K • ) and End •Aλ (Kλ• ) are quasiisomorphic
graded DG-algebras.
We already saw that A being a Koszul ring implies that
End •A (K • ) ∼
= (Ext •A (k, k), d = 0)
and by the above and since Aλ is Koszul as well we have that
End •Aλ (T0λ K • ) ∼
= (Ext •Aλ (T0λ k, T0λ k), d = 0) ∼
= (Ext •Aλ (kλ , kλ ), d = 0)
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Ryom-Hansen
We now obtain the following commutative diagram of functors:
Db (mod−A)
↓ T0λ
Db (mod−Aλ )
− ⊗ K•
←
− ⊗ T0λ K •
←
Db (hEnd •A (K • )i)
↓
Db (hExt •A (k, k), d = 0i)
↓ T0λ
←
Db (hEnd •Aλ (T0λ K • )i) ← Db (hExt •Aλ (T0λ k, T0λ k), d = 0i)
where the middle vertical arrow is the graded tensor product functor
− ⊗End •A (K • ) End •A (T0λ K • )
while the first and third vertical arrows are the graded translation functors. The
commutativity of the first square is here obvious, whereas the quasiisomorphism
of (Ext •A (k, k), d = 0)) × End •Aλ (T0λ K • )-bimodules
End •A (T0λ K • ) ∼
= Ext •Aλ (T0λ k, T0λ k)
gives the natural transformation that makes the second square commutative.
By the same reasoning as in the last section, the two lower arrows define
equivalences of triangulated categories.
Now Backelin [2] shows that one can choose the isomorphisms Ext •A (k, k) '
A and Ext •Aλ (T0λ k, T0λ k) ' Aλ to obtain the following commutative diagram:
∼
Ext •A (k, k)
−→
T0λ ↓
∼
Ext •Aλ (T0λ k, T0λ k) −→
A
↓τ
Aλ
The key point is here that both vertical maps are surjections: for τ this is clear,
while for T0λ an argument involving the Koszul property of Ais required (one can
here give a simple alternate argument along the lines of the Cline, Parshall, Scott
approach to Kashdan-Luzstig theory [8]). It then follows that the kernels of the
two vertical maps are the ideals generated by corresponding idempotents (thus in
the degree 0 part).
Although the diagram involves non-graded maps, it can be used to give τ
a grading – and then of course it is a commutative diagram of graded homomorphisms.
This diagram, on the other hand, gives rise to the following commutative
diagram of functors:
F
Db (hExt •A (k, k), d = 0i)
← Db (mod−A)
T0λ ↓
↓τ
F
Db (hExt •Aλ (T0λ k, T0λ k), d = 0i) ← Db (mod−Aλ )
We now join the two diagrams of functors to obtain a diagram, in which the
upper arrows compose to the Koszul duality functor of O while the composition
of the lower arrows is isomorphic to the Koszul duality functor of Oλ . Let us
formulate this as a Theorem
Ryom-Hansen
161
Theorem 4.1.
The translation– and Zuckerman functors are Kozsul to each
other, in other words there is a commutative diagram of functors:
Db (mod−A)
T0λ ↓
D
←
Db (mod−A)
↓τ
D
Db (mod−Aλ ) ←λ Db (mod−Aλ )
where D (resp. Dλ ) is the Koszul duality functor as described above.
This is the Theorem announced in the introduction of the paper.
5.
The Enright-Shelton Equivalence
We now consider the categorification of the Temperley-Lieb algebra. Let thus
g := gln and O := O(gln ). Let k , k ∈ {1, 2, . . . , n} be the standard basis of the
weight lattice and define for k ∈ {1, 2, . . . , n}, λk := 1 + 2 + . . . k . We then let
Ok,n−k be the singular block of O consisting of modules with central character
θ(λk ). So the Verma module with highest weight λk − ρ lies in Ok,n−k . Let
gi , 1 ≤ i ≤ i − 1 be the subalgebra of g consisting of the matrices whose entries
are nonzero only on the intersection of the i-th and (i + 1)-th rows and columns.
i
We then denote by Ok,n−k
the parabolic subcategory of Ok,n−k whose modules are
the locally gi -finite ones in Ok,n−k .
We shall also consider the following dual picture: let pk be the parabolic
subalgebra of g, whose Levi part is gk ⊕gn−k and such that n+ ⊆ pk and let Ok,n−k
be the the full subcategory of O consisting of the locally pk -finite modules. Choose
an integral dominant regular weight µ and integral dominant subregular weights
µi on the i-th wall, i = 1, 2, . . . , n − 1 (so the coordinates of µi in the i basis
are strictly decreasing, except for the i-th and the (i + 1)-th that are equal). Let
finally Oik,n−k be the subcategory of Ok,n−k with central character θ(µi ).
All of this is the setup of [6]
i
Let Rk,n−k
(resp.Rik,n−k ) be the endomorphism ring of the minimal proi
(resp. Oik,n−k ). The following theorem is a direct
jective generator of Ok,n−k
consequence of Backelin’s work [2]:
Theorem 5.1.
Proof.
i
(Rk,n−k
)! = Rik,n−k
The main result of [2] is that
φ
(Rφψ )! = R−w
0ψ
with the notation as in [5].
Now one
Pshould first observe that c Id ∈ gln acts on Oλ through multiplication with c λi ; hence Oλ is equivalent to the category Oλ of sln -modules,
where λ P
denotes the image of λ under the projection of the weight lattice with
kernel Z i . We can thus restrict ourselves to the semisimple situation and may
indeed use the results quoted.
Now µi and sli ⊆ sln are both given by the simple root αi = i − i+1 and
also pk,n−k and λk are given by the same simple roots (= ∆ \ {αi }). Finally, we
have in type A that w0 λ = −λ and the theorem follows.
162
Ryom-Hansen
As a corollary we obtain a simple proof of the following equivalence of
categories first proven by Enright and Shelton.
Corollary 5.2.
Proof.
O1k,n−k ∼
= Ok−1,n−k−1
There is first of all a standard equivalence of categories:
1
∼
Ok,n−k
= Ok−1,n−k−1
The functor νn from left to right takes the sum of all weight spaces of weight
1 + x3 3 + . . . xn n − ρn with xi ∈ Z. This is a gln−2 -module and νn is then the
tensor product of it with the module defined by 1 + 2 + . . . n−2 . The inverse
functor comes from an induction procedure, see [6] for the details. Now this
equivalence gives us a ring isomorphism
1
(Rk,n−k
)! ∼
= (Rk−1,n−k−1 )!
which combined with the theorem yields a ring isomorphism
ξ : R1k,n−k ∼
= Rk−1,n−k−1
But then the module categories of R1k,n−k and Rk−1,n−k−1 are equivalent, by the
restriction and extension of scalars along ξ . The corollary is proved.
This might be useful in proving the full conjectures of [6].
References
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[3]
[4]
[5]
[6]
[7]
[8]
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[10]
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Steen Ryom-Hansen
Matematisk Afdeling
Universitetsparken 5
DK-2100 København Ø
Danmark
steen@math.ku.dk
Received October 16, 2002
and in final form October 13, 2003